Robust Semantic Fragmentation and Proof Width: A Topological–PCP Approach to Lower Bounds in Proof Complexity

This paper develops a robust notion of semantic fragmentation designed to survive known obstructions in global-structure approaches to P vs NP. Building on earlier results showing that naive Helly-type invariants collapse under polynomial-time reductions, it introduces a restriction-stable refinement and proves that robust semantic fragmentation implies linear lower bounds on resolution width. Explicit EXACTLY-ONE surface formulas are shown to satisfy this robust property, yielding concrete proof-complexity lower bounds. The paper analyzes how robustness behaves under reductions, identifies locality-preserving reductions as the correct invariant class, and reduces further progress to the construction of robust PCP systems. The framework clarifies the precise technical bottlenecks separating semantic hardness from unconditional complexity separations.